36 ideas
| 23250 | Desired responsible actions result either from rational or from irrational desire [Aristotle] |
| Full Idea: And of responsible actions, some are done through habit, some through desire, and of these some through rational and some through irrational desire. | |
| From: Aristotle (The Art of Rhetoric [c.350 BCE], 1369a01) | |
| A reaction: Identified by Michael Frede, to illustrate reason having its own distinctive type of desire ('Boulesis'). I suspect that the rational desires are the morally good desires. |
| 5847 | It is the role of dialectic to survey syllogisms [Aristotle] |
| Full Idea: It belongs to dialectic to survey equally all kinds of syllogism. | |
| From: Aristotle (The Art of Rhetoric [c.350 BCE], 1355a08) | |
| A reaction: Since dialectic is central to philosophy, this implies that philosophers ought to be students of logic. This duty seems to me to be taken more seriously in the analytical tradition than in the 'continental' tradition. |
| 10170 | While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price] |
| Full Idea: While truth can be defined in a relative way, as truth in one particular model, a non-relative notion of truth is implied, as truth in all models. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: [The article is actually discussing arithmetic] This idea strikes me as extremely important. True-in-all-models is usually taken to be tautological, but it does seem to give a more universal notion of truth. See semantic truth, Tarski, Davidson etc etc. |
| 10166 | ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price] |
| Full Idea: In standard ZFC ('Zermelo-Fraenkel with Choice') set theory we deal merely with pure sets, not with additional urelements. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: The 'urelements' would the actual objects that are members of the sets, be they physical or abstract. This idea is crucial to understanding philosophy of mathematics, and especially logicism. Must the sets exist, just as the urelements do? |
| 10175 | Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price] |
| Full Idea: In second-order logic there are three kinds of variables, for objects, for functions, and for predicates or sets. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: It is interesting that a predicate seems to be the same as a set, which begs rather a lot of questions. For those who dislike second-order logic, there seems nothing instrinsically wicked in having variables ranging over innumerable multi-order types. |
| 10165 | 'Analysis' is the theory of the real numbers [Reck/Price] |
| Full Idea: 'Analysis' is the theory of the real numbers. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: 'Analysis' began with the infinitesimal calculus, which later built on the concept of 'limit'. A continuum of numbers seems to be required to make that work. |
| 10174 | Mereological arithmetic needs infinite objects, and function definitions [Reck/Price] |
| Full Idea: The difficulties for a nominalistic mereological approach to arithmetic is that an infinity of physical objects are needed (space-time points? strokes?), and it must define functions, such as 'successor'. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: Many ontologically austere accounts of arithmetic are faced with the problem of infinity. The obvious non-platonist response seems to be a modal or if-then approach. To postulate infinite abstract or physical entities so that we can add 3 and 2 is mad. |
| 10164 | Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price] |
| Full Idea: A common formulation of Peano Arithmetic uses 2nd-order logic, the constant '1', and a one-place function 's' ('successor'). Three axioms then give '1 is not a successor', 'different numbers have different successors', and induction. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: This is 'second-order' Peano Arithmetic, though it is at least as common to formulate in first-order terms (only quantifying over objects, not over properties - as is done here in the induction axiom). I like the use of '1' as basic instead of '0'! |
| 10172 | Set-theory gives a unified and an explicit basis for mathematics [Reck/Price] |
| Full Idea: The merits of basing an account of mathematics on set theory are that it allows for a comprehensive unified treatment of many otherwise separate branches of mathematics, and that all assumption, including existence, are explicit in the axioms. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: I am forming the impression that set-theory provides one rather good model (maybe the best available) for mathematics, but that doesn't mean that mathematics is set-theory. The best map of a landscape isn't a landscape. |
| 10167 | Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price] |
| Full Idea: Structuralism has emerged from the development of abstract algebra (such as group theory), the creation of axiom systems, the introduction of set theory, and Bourbaki's encyclopaedic survey of set theoretic structures. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §2) | |
| A reaction: In other words, mathematics has gradually risen from one level of abstraction to the next, so that mathematical entities like points and numbers receive less and less attention, with relationships becoming more prominent. |
| 10169 | Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price] |
| Full Idea: Relativist Structuralism simply picks one particular model of axiomatised arithmetic (i.e. one particular interpretation that satisfies the axioms), and then stipulates what the elements, functions and quantifiers refer to. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: The point is that a successful model can be offered, and it doesn't matter which one, like having any sort of aeroplane, as long as it flies. I don't find this approach congenial, though having a model is good. What is the essence of flight? |
| 10179 | There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price] |
| Full Idea: The term 'structure' has two uses in the literature, what can be called 'particular structures' (which are particular relational systems), but also what can be called 'universal structures' - what particular systems share, or what they instantiate. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §6) | |
| A reaction: This is a very helpful distinction, because it clarifies why (rather to my surprise) some structuralists turn out to be platonists in a new guise. Personal my interest in structuralism has been anti-platonist from the start. |
| 10181 | Pattern Structuralism studies what isomorphic arithmetic models have in common [Reck/Price] |
| Full Idea: According to 'pattern' structuralism, what we study are not the various particular isomorphic models of arithmetic, but something in addition to them: a corresponding pattern. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §7) | |
| A reaction: Put like that, we have to feel a temptation to wield Ockham's Razor. It's bad enough trying to give the structure of all the isomorphic models, without seeking an even more abstract account of underlying patterns. But patterns connect to minds.. |
| 10182 | There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price] |
| Full Idea: There are four main variants of structuralism in the philosophy of mathematics - formalist structuralism, relativist structuralism, universalist structuralism (with modal variants), and pattern structuralism. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §9) | |
| A reaction: I'm not sure where Chihara's later book fits into this, though it is at the nominalist end of the spectrum. Shapiro and Resnik do patterns (the latter more loosely); Hellman does modal universalism; Quine does the relativist version. Dedekind? |
| 10168 | Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price] |
| Full Idea: Formalist Structuralism endorses structural methodology in mathematics, but rejects semantic and metaphysical problems as either meaningless, or purely formal, or as inference relations. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §3) | |
| A reaction: [very compressed] I find the third option fairly congenial, certainly in preference to rather platonist accounts of structuralism. One still needs to distinguish the mathematical from the non-mathematical in the inference relations. |
| 10178 | Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price] |
| Full Idea: It is tempting to take a modal turn, and quantify over all possible objects, because if there are only a finite number of actual objects, then there are no models (of the right sort) for Peano Arithmetic, and arithmetic is vacuously true. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: [compressed; Geoffrey Hellman is the chief champion of this view] The article asks whether we are not still left with the puzzle of whether infinitely many objects are possible, instead of existent. |
| 10176 | Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price] |
| Full Idea: Universalist Structuralism is a semantic thesis, that an arithmetical statement asserts a universal if-then statement. We build an if-then statement (using quantifiers) into the structure, and we generalise away from any one particular model. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: There remains the question of what is distinctively mathematical about the highly generalised network of inferences that is being described. Presumable the axioms capture that, but why those particular axioms? Russell is cited as an originator. |
| 10177 | Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price] |
| Full Idea: Universalist Structuralism is eliminativist about abstract objects, in a distinctive form. Instead of treating the base element (say '1') as an ambiguous referring expression (the Relativist approach), it is a variable which is quantified out. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §5) | |
| A reaction: I am a temperamental eliminativist on this front (and most others) so this is tempting. I am also in love with the concept of a 'variable', which I take to be utterly fundamental to all conceptual thought, even in animals, and not just a trick of algebra. |
| 10171 | The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price] |
| Full Idea: Relativist Structuralism must first assume the existence of an infinite set, otherwise there would be no model to pick, and arithmetical terms would have no reference. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: See Idea 10169 for Relativist Structuralism. They point out that ZFC has an Axiom of Infinity. |
| 10173 | A nominalist might avoid abstract objects by just appealing to mereological sums [Reck/Price] |
| Full Idea: One way for a nominalist to reject appeal to all abstract objects, including sets, is to only appeal to nominalistically acceptable objects, including mereological sums. | |
| From: E Reck / M Price (Structures and Structuralism in Phil of Maths [2000], §4) | |
| A reaction: I'm suddenly thinking that this looks very interesting and might be the way to go. The issue seems to be whether mereological sums should be seen as constrained by nature, or whether they are unrestricted. See Mereology in Ontology...|Intrinsic Identity. |
| 5862 | A single counterexample is enough to prove that a truth is not necessary [Aristotle] |
| Full Idea: If we have a single counter-instance, the argument is refuted as not necessary, even if more cases are otherwise or more often otherwise. | |
| From: Aristotle (The Art of Rhetoric [c.350 BCE], 1403a07) | |
| A reaction: This is Aristotle (pioneering hero) pointing out what we now tend to think of as Karl Popper's falsification, the certain way to demonstrate the falseness of a supposed law of nature, by finding one anomaly from it. |
| 5854 | Nobody fears a disease which nobody has yet caught [Aristotle] |
| Full Idea: Nobody is on his guard against a disease that nobody has yet caught. | |
| From: Aristotle (The Art of Rhetoric [c.350 BCE], 1372a27) | |
| A reaction: A beautifully simple indication of one problem with induction. In a dangerous situation, you can't wait around for a few experiences in order to learn the regularities and rules. Either you are doomed, or you must explain using related experiences. |
| 5849 | Rhetoric is a political offshoot of dialectic and ethics [Aristotle] |
| Full Idea: Rhetoric is a kind of offshoot of dialectic and of the study of ethics, and is quite properly categorized as political. | |
| From: Aristotle (The Art of Rhetoric [c.350 BCE], 1356a25) | |
| A reaction: Aristotle gives a higher status to rhetoric than Socrates and Plato did - and rightly, in my view. We have lost sight of it as a vital part of politics, and philosophers must fight for virtue in rhetoric, which requires right reason and fine principles. |
| 5851 | Pentathletes look the most beautiful, because they combine speed and strength [Aristotle] |
| Full Idea: The pentathletes are the most beautiful, being at the same time naturally suited to both speed and force. | |
| From: Aristotle (The Art of Rhetoric [c.350 BCE], 1361b09) | |
| A reaction: This is still true. Watch the Olympics. The bodies we envy most belong to those who do a variety of disciplines. The most beautiful music fulfils a variety of functions (structure, as well as melody, drama, rhythm, harmony, novelty). |
| 5858 | Men are physically prime at thirty-five, and mentally prime at forty-nine [Aristotle] |
| Full Idea: The body is in its prime from the ages of thirty to thirty-five, and the soul around the age of forty-nine. | |
| From: Aristotle (The Art of Rhetoric [c.350 BCE], 1390b09) | |
| A reaction: Wonderfully specific! It is important that Aristotle is interested in these questions. The good for man follows the path laid out by nature, in which a man rises to his highest good in maturity, and then declines from it into old age. |
| 5855 | We all feel universal right and wrong, independent of any community or contracts [Aristotle] |
| Full Idea: There is something of which we all have an inkling, being a naturally universal right and wrong, even if there should be no community between the parties or contract. | |
| From: Aristotle (The Art of Rhetoric [c.350 BCE], 1373b07) | |
| A reaction: This is the strongest assertion I know of in Aristotle of an absolute moral standard, independent of natural function. It makes Aristotle an intuitionist, and is strikingly opposed to contracts as the most basic aspect of morality. |
| 5850 | Happiness is composed of a catalogue of internal and external benefits [Aristotle] |
| Full Idea: The elements of happiness are: gentle birth, many virtuous friends, wealth, creditable and extensive offspring, a comfortable old age; also health, beauty, strength, size and competitiveness, reputation, status, luck and the virtues. | |
| From: Aristotle (The Art of Rhetoric [c.350 BCE], 1360b18) | |
| A reaction: This is Aristotle's pluralism, and his commitment to 'external goods' (rather than the inner good of pure virtue, which the Stoics preferred). 'Gentle birth' might turn out to mean good upbringing and education. Who was the most 'beautiful' philosopher? |
| 5856 | Self-interest is a relative good, but nobility an absolute good [Aristotle] |
| Full Idea: One's own interest is a relative good, nobility a good absolutely. | |
| From: Aristotle (The Art of Rhetoric [c.350 BCE], 1389b37) | |
| A reaction: The key idea in the whole of Greek moral theory is the concept of what we can call a 'beautiful' action. Such things, or course, tend to be visible in great actions, such as sparing an enemy, rather than the minutiae of well-mannered daily life. |
| 5848 | All good things can be misused, except virtue [Aristotle] |
| Full Idea: If one used strength, health, wealth and strategic expertise well, one might do the greatest possible good and if badly the greatest possible harm; this is a problem common to all good things, except virtue. | |
| From: Aristotle (The Art of Rhetoric [c.350 BCE], 1355b04) | |
| A reaction: Of course, this may just be a tautology about virtue, rather than an empirical observation. However, in 'Ethics' he tries to describe a state of mind (essentially one of harmony) which could never result in misuse. |
| 5853 | The best virtues are the most useful to others [Aristotle] |
| Full Idea: The greatest virtues must be those most useful to others. | |
| From: Aristotle (The Art of Rhetoric [c.350 BCE], 1366b02) | |
| A reaction: I wonder if this applies to the intellectual virtues, as well as to the social virtues? Is this virtue theory's answer to utilitarianism, or utilitarianism's answer to virtue theory? Personally I think good persons are prior to benefits. |
| 5857 | The young feel pity from philanthropy, but the old from self-concern [Aristotle] |
| Full Idea: Old men are prone to pity, but where the young are so from philanthropy, the old are so from weakness, for they think all these things are near for themselves to suffer. | |
| From: Aristotle (The Art of Rhetoric [c.350 BCE], 1390a18) | |
| A reaction: I am shocked to find Aristotle being so cynical. I see no reason why the old should not be as philanthropic as anyone else, and they clearly are so, as when they plant trees for future generations to enjoy. |
| 5859 | Rich people are mindlessly happy [Aristotle] |
| Full Idea: The character of the rich man is that of the mindlessly happy one. | |
| From: Aristotle (The Art of Rhetoric [c.350 BCE], 1391a12) | |
| A reaction: Very nice. It is hard to deny that the rich tend to be happy (in some sense of the word), and recent sociological research has tended to demonstrate this, but the pursuit of wealth must inevitably take the focus away from key intellectual pursuits. Yeh? |
| 20767 | Culture is now dominated by boredom, so universal it is unnoticed [Heidegger, by Aho] |
| Full Idea: Heidegger came to say that the cultural mood had changed from 'anxiety' to 'boredom'. The danger is that our boredom has become so ubiquitous and all-encompassing that it is now hidden. | |
| From: report of Martin Heidegger (Contributions to Philosophy [1938]) by Kevin Aho - Existentialism: an introduction 9 'Conc' | |
| A reaction: I'm not sure what the danger of boredom is if it is 'hidden'. It rather depends what else is hidden with it. |
| 5852 | The four constitutions are democracy (freedom), oligarchy (wealth), aristocracy (custom), tyranny (security) [Aristotle] |
| Full Idea: There are four types of constitution: democracy (whose purpose is freedom), oligarchy (for wealth), aristocracy (for education and customs), and monarch or tyranny (for security). | |
| From: Aristotle (The Art of Rhetoric [c.350 BCE], 1365b28-37) | |
| A reaction: An aristocracy seems to be the guardians of tradition and culture (as in an English public school education). The tyranny of Hitler and Stalin did not exactly lead to security. Democracy and aristocracy are the front-runners. Compare Idea 2821. |
| 1660 | It is noble to avenge oneself on one's enemies, and not come to terms with them [Aristotle] |
| Full Idea: It is noble to avenge oneself on one's enemies and not to come to terms with them. | |
| From: Aristotle (The Art of Rhetoric [c.350 BCE], 1367a19), quoted by Gregory Vlastos - Socrates: Ironist and Moral Philosopher p.189 |
| 5861 | People assume events cause what follows them [Aristotle] |
| Full Idea: Men take its occurring after as its occurring because. | |
| From: Aristotle (The Art of Rhetoric [c.350 BCE], 1401b30) | |
| A reaction: The Latin is 'post hoc propter hoc' - after this so because of this. It is quite a good inductive rule, but obviously open to abuse, as in legal cases, as when someone happens to acquire a lot of money just after a crime. |